Together with Giovanni, we have now exchanged several personal emails and finally resolved my problems. I was successfully able to program the HC-05 interface with my Raspberry PI 3 (newest Raspbian version).
The steps were (the following text is mostly from Giovanni):
1) Raspberry PI 3 uses ttyAMA0 for Bluetooth, so you have to use
2) The serial line must be enabled in /boot/config.txt using the line
3) due to the limitation of the UART used, that relies on CPU clock,
in /boot/config.txt the cpu speed has to be fixed using the line
4) In the graphical configuration (Menu "Settings" - "Raspberry PI configuration") in the "Interfaces" tab (or in the terminal with sudo raspi-config), the "Serial" interface may be enabled (or not), this does apparently not harm.
5) To avoid the occupation of the serial line by a console you have to
verify in /boot/cmdline.txt that the parameter console has no
reference to things like serial o dev but only to tty1 or similar
6) In case of an already present errno 11, it could be of help to try
to stop login consoles on /dev/ttyS0 using the command
sudo systemctl stop serial.getty@...
So, I did not need to buy a TTL interface, all worked well with the Raspberry PI.
Maybe this helps to other OMs who want also make their own HC-05 with a new Raspberry PI 3.
After connecting the HC05 to a 9-pin-plug, downloading the app and updating the DUO firmware, now all works very well.
Many thanks to Giovanni for the huge help he provided.